Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter VII

by Josiah Willard Gibbs

VII.

ON THE RÔLE OF QUATERNIONS IN THE ALGEBRA OF VECTORS.

[Nature, vol. xliii. pp. 511–513, April 2, 1891.]

The following passage, which has recently come to my notice, in the preface to the third edition of Prof. Tait's Quaternions seems to call for some reply:

"Even Prof. Willard Gibbs must be ranked as one of the retarders of quaternion progress, in virtue of his pamphlet on Vector Analysis, a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann."

The merits or demerits of a pamphlet printed for private distribution a good many years ago do not constitute a subject of any great importance, but the assumptions implied in the sentence quoted are suggestive of certain reflections and inquiries which are of broader interest, and seem not untimely at a period when the methods and results of the various forms of multiple algebra are attracting so much attention. It seems to be assumed that a departure from quaternionic usage in the treatment of vectors is an enormity. If this assumption is true, it is an important truth; if not, it would be unfortunate if it should remain unchallenged, especiaUy when supported by so high an authority. The criticism relates particularly to notations, but I believe that there is a deeper question of notions underlying that of notations. Indeed, if my offence had been solely in the matter of notation, it would have been less accurate to describe my production as a monstrosity, than to characterize its dress as uncouth.

Now what are the fundamental notions which are germane to a vector analysis? (A vector analysis is of course an algebra for vectors, or something which shall be to vectors what ordinary algebra is to ordinary quantities.) If we pass over those notions which are so simple that they go without saying, geometrical addition (denoted by $+$) is, perhaps, first to be mentioned. Then comes the product of the lengths of two vectors and the cosine of the angle which they include. This, taken negatively, is denoted in quaternions by ${\text{S}}\alpha \beta$, where $\alpha$ and $\beta$ are the vectors. Equally important is a vector at right angles to $\alpha$ and $\beta$ (on a specified side of their plane), and representing in length the product of their lengths and the sine of the angle which they include. This is denoted by in quaternions. How these notions are represented in my pamphlet is a question of very subordinate consequence, which need not be considered at present. The importance of these notions, and the importance of a suitable notation for them, is not, I suppose, a matter on which there is any difference of opinion. Another function of $\alpha$ and $\beta ,$ called their product and written $\alpha \beta ,$ is used in quaternions. In the general case, this is neither a vector, like ${\text{V}}\alpha \beta ,$ nor a scalar (or ordinary algebraic quantity), like ${\text{S}}\alpha \beta$ but a quaternion—that is, it is part vector and part scalar. It may be defined by the equation—

$\alpha \beta ={\text{V}}\alpha \beta +{\text{S}}\alpha \beta .$

The question arises, whether the quatemionic product can claim a prominent and fundamental place in a system of vector analysis. It certainly does not hold any such place among the fundamental geometrical conceptions as the geometrical sum, the scalar product, or the vector product. The geometrical sum $\alpha +\beta$ represents the third side of a triangle as determined by the sides $\alpha$ and $\beta$ ${\text{V}}\alpha \beta$ represents in magnitude the area of the parallelogram determined by the sides $\alpha$ and $\beta ,$ and in direction the normal to the plane of the parallelogram. ${\text{S}}\gamma {\text{V}}\alpha \beta$ represents the volume of the parallelopiped determined by the edges $\alpha ,\beta ,$ and $\gamma .$ These conceptions are the very foundations of geometry.

We may arrive at the same conclusion from a somewhat narrower but very practical point of view. It will hardly be denied that sines and cosines play the leading parts in trigonometry. Now the notations ${\text{V}}\alpha \beta$ and ${\text{S}}\alpha \beta$ represent the sine and the cosine of the angle included between $\alpha$ and $\beta ,$ combined in each case with certain other simple notions. But the sine and cosine combined with these auxiliary notions are incomparably more amenable to analytical transformation than the simple sine and cosine of trigonometry, exactly as numerical quantities combined (as in algebra) with the notion of positive or negative quality are incomparably more amenable to analytical transformation than the simple numerical quantities of arithmetic.

I do not know of anything which can be urged in favor of the quaternionic product of two vectors as a fundamental notion in vector analysis, which does not appear trivial or artificial in comparison with the above considerations. The same is true of the quatemionic quotient, and of the quaternion in general.

How much more deeply rooted in the nature of things are the functions ${\text{V}}\alpha \beta$ and ${\text{S}}\alpha \beta$ than any which depend on the definition of a quaternion, will appear in a strong light if we try to extend our formulæ to space of four or more dimensions. It will not be claimed that the notions of quaternions will apply to such a space, except indeed in such a limited and artificial manner as to rob them of their value as a system of geometrical algebra. But vectors exist in such a space, and there must be a vector analysis for such a space. The notions of geometrical addition and the scalar product are evidently applicable to such a space. As we cannot define the direction of a vector in space of four or more dimensions by the condition of perpendicularity to two given vectors, the definition of ${\text{V}}\alpha \beta$, as given above, will not apply totidem verbis to space of four or more dimensions. But a little change in the definition, which would make no essential difierence in three dimensions, would enable us to apply the idea at once to space of any number of dimensions.

These considerations are of a somewhat a priori nature. It may be more convincing to consider the use actually made of the quaternion as an instrument for the expression of spatial relations. The principal use seems to be the derivation of the functions expressed by ${\text{S}}\alpha \beta$ and ${\text{V}}\alpha \beta .$ Each of these expressions is regarded by quatemionic writers as representing two distinct operations; first, the formation of the product $\alpha \beta ,$ which is the quaternion, and then the taking out of this quaternion the scalar or the vector part, as the case may be, this second process being represented by the selective symbol, ${\text{S}}$ or ${\text{V.}}$ This is, I suppose, the natural development of the subject in a treatise on quaternions, where the chosen subject seems to require that we should commence with the idea of a quaternion, or get there as soon as possible, and then develop everything from that particular point of view. In a system of vector analysis, in which the principle of development is not thus predetermined, it seems to me contrary to good method that the more simple and elementary notions should be defined by means of those which are less so.

The quaternion affords a convenient notation for rotations. The notation $q(\,\,)q^{-1},$ where $q$ is a quaternion and the operand is to be written in the parenthesis, produces on all possible vectors just such changes as a (finite) rotation of a solid body. Rotations may also be represented, in a manner which seems to leave nothing to be desired, by linear vector functions. Doubtless each method has advantages in certain cases, or for certain purposes. But since nothing is more simple than the definition of a linear vector function, while the definition of a quaternion is far from simple, and since in any case linear vector functions must be treated in a system of vector analysis, capacity for representing rotations does not seem to me sufficient to entitle the quaternion to a place among the fundamental and necessary notions of a vector analysis. Another use of the quaternionic idea is associated with the symbol $\nabla .$ The quantities written ${\text{S}}\nabla \omega$ and ${\text{V}}\nabla \omega$ where $\omega$ denotes a vector having values which vary in space, are of fundamental importance in physics. In quaternions these are derived from the quaternion $\nabla \omega$ by selecting respectively the scalar or the vector part. But the most simple and elementary definitions of ${\text{S}}\nabla \omega$ and ${\text{V}}\nabla \omega$ are quite independent of the conception of a quaternion, and the quaternion $\nabla \omega$ is scarcely used except in combination with the symbols ${\text{S}}$ and ${\text{V}},$ expressed or implied. There are a few formulæ in which there is a trifling gain in compactness in the use of the quaternion, but the gain is very trifling so far as I have observed, and generally, it seems to me, at the expense of perspicuity.

These considerations are sufficient, I think, to show that the position of the quatemionist is not the only one from which the subject of vector analysis may be viewed, and that a method which would be monstrous from one point of view, may be normal and inevitable from another.

Let us now pass to the subject of notations. I do not know wherein the notations of my pamphlet have any special resemblance to Grassmann's, although the point of view from which the pamphlet was written is certainly much nearer to his than to Hamilton's. But this a matter of minor consequence. It is more important to ask. What are the requisites of a good notation for the purposes of vector analysis? There is no difference of opinion about the representation of geometrical addition. When we come to functions having an analogy to multiplication, the products of the lengths of two vectors and the cosine of the angle which they include, from any point of view except that of the quatemionist, seems more simple than the same quantity taken negatively. Therefore we want a notation for what is expressed by $-{\text{S}}\alpha \beta ,$ rather than ${\text{S}}\alpha \beta ,$ in quaternions. Shall the symbol denoting this function be a letter or some other sign? and shall it precede the vectors or be placed between them? A little reflection will show, I think, that while we must often have recourse to letters to supplement the number of signs available for the expression of all kinds of operations, it is better that the symbols expressing the most fundamental and frequently recurring operations should not be letters, and that a sign between the vectors, and, as it were, uniting them, is better than a sign before them in a case having a formal analogy with multiplication. The case may be compared with that of addition, for which $\alpha +\beta$ is evidently more convenient than $\textstyle \sum \displaystyle (\alpha ,\beta )$ or $\textstyle \sum \displaystyle \alpha \beta$ would be. Similar considerations will apply to the function written in quaternions ${\text{V}}\alpha \beta .$ It would seem that we obtain the ne plus ultra of simplicity and convenience, if we express the two functions by uniting the vectors in each case with a sign suggestive of multiplication. The particular forms of the signs which we adopt is a matter of minor consequence. In order to keep within the resources of an ordinary printing office, I have used a dot and a cross, which are abeady associated with multiplication, but are not needed for ordinary multiplication, which is best denoted by the simple juxtaposition of the factors. I have no especial predilection for these particular signs. The use of the dot is indeed liable to the objection that it interferes with its use as a separatrix, or instead of a parenthesis.

If, then, I have written $\alpha .\beta$ and $\alpha \times \beta$ for what is expressed in quaternions by $-{\text{S}}\alpha \beta$ and ${\text{V}}\alpha \beta ,$ and in like manner $\nabla .\omega$ and $\nabla \times \omega$ for $-{\text{S}}\nabla \omega$ and ${\text{V}}\nabla \omega$ in quaternions, it is because the natural development of a vector analysis seemed to lead logically to some such notations. But I think that I can show that these notations have some substantial advantages over the quatemionic in point of convenience.

Any linear vector function of a variable vector $\rho$ may be expressed in the form—

$\alpha \lambda .\rho +\beta \mu .\rho +\gamma \nu .\rho =(\alpha \lambda +\beta \mu +\gamma \nu ).\rho =-\Phi .\rho ,$

where

$\Phi =\alpha \lambda +\beta \mu +\gamma \nu ;$

or in quaternions

$-\alpha {\text{S}}\lambda \rho -\beta {\text{S}}\mu \rho -\gamma {\text{S}}\nu \rho =-(\alpha {\text{S}}\lambda +\beta {\text{S}}\mu +\gamma {\text{S}}\nu )\rho =-\phi \rho ,$

where

$\phi =\alpha {\text{S}}\lambda +\beta {\text{S}}\mu +\gamma {\text{S}}\nu .$

If we take the scalar product of the vector $\Phi .\rho ,$ and another vector $\sigma ,$ we obtain the scalar quantity

$\sigma .\Phi .\rho =\sigma .(\alpha \lambda +\beta \mu +\gamma \nu ).\rho ,$

or in quaternions

${\text{S}}\sigma \phi \rho ={\text{S}}\sigma (\alpha {\text{S}}\lambda +\beta {\text{S}}\mu +\gamma {\text{S}}\nu )\rho .$

This is a function of $\sigma$ and of $\rho ,$ and it is exactly the same kind of function of $\sigma$ that it is of $\rho ,$ a symmetry which is not so clearly exhibited in the quaternionic notation as in the other. Moreover, we can write $\sigma .\Phi$ for $\sigma .(\alpha \lambda +\beta \mu +\gamma \nu ).$ This represents a vector which is a function of $\sigma ,$ viz., the function conjugate to $\Phi .\sigma ;$ and $\sigma .\Phi .\rho$ may be regarded as the product of this vector and $\rho .$ This is not so clearly indicated in the quaternionic notation, where it would be straining things a little to call ${\text{S}}\sigma \phi$ a vector.

The combinations $\alpha \lambda ,\beta \mu ,$ etc., used above, are distributive with regard to each of the two vectors, and may be regarded as a kind of product. If we wish to express everything in terms of $i,j,$ and $k,$ $\Phi$ will appear as a sum of ${ii,ij,ik,ji,jj,jk,ki,kj,kk,}$ each with a numerical coefficient. These nine coefficients may be arranged in a square, and constitute a matrix; and the study of the properties of expressions like $\Phi$ is identical with the study of ternary matrices. This expression of the matrix as a sum of products (which may be extended to matrices of any order) affords a point of departure from which the properties of matrices may be deduced with the utmost facility. The ordinary matricular product is expressed by a dot, as $\Phi .\Psi .$. Other important kinds of multiplication may be defined by the equations—

$(\alpha \lambda )_{\times }^{\times }(\beta \mu )=(\alpha \times \beta )(\lambda \times \mu ),$$(\alpha \lambda ):(\beta \mu )=(\alpha .\beta )(\lambda .\mu ).$

With these definitions ${\tfrac {1}{6}}\Phi _{\times }^{\times }\Phi :\Phi$ will be the determinant of $\Phi ,$ and $\Phi _{\times }^{\times }\Phi$ will be the conjugate of the reciprocal of $\Phi$ multiplied by twice the determinant. If $\Phi$ represents the manner in which vectors are affected by a strain, ${\tfrac {1}{2}}\Phi _{\times }^{\times }\Phi$ will represent the manner in which surfaces are affected, and ${\tfrac {1}{6}}\Phi _{\times }^{\times }\Phi :\Phi$ the manner in which volumes are affected. Considerations of this kind do not attach themselves so naturally to the notation $\phi =\alpha {\text{S}}\lambda +\beta {\text{S}}\mu +\gamma {\text{S}}\nu ,$ nor does the subject admit so free a development with this notation, principally because the symbol ${\text{S}}$ refers to a special use of the matrix, and is very much in the way when we want to apply the matrix to other uses, or to subject it to various operations.